Geodesic
Polynomial deformations of the Lie algebras $sl(2)$ in problems of quantum optics
Teoretičeskaâ i matematičeskaâ fizika, Tome 95 (1993) no. 1, pp. 3-19 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is shown that specific (polynomial) deformations of Lie algebras arise naturally as dynamical symmetry algebras $g^{ds}$ of second-quantized models with nonquadratic Hamiltonians $H$ invariant with respect to certain groups $G^{\text {inv}}(H)$. Such deformations $sl_ d(2)$ of the Lie algebras $sl(2)$ are found in a number of models of quantum optics (multiphoton processes, generalized Dicke model, and frequency conversion), and ways to apply thes $sl(2)$ formalism to the solution of physics problems are indicated. 
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V. P. Karassiov. Polynomial deformations of the Lie algebras $sl(2)$ in problems of quantum optics. Teoretičeskaâ i matematičeskaâ fizika, Tome 95 (1993) no. 1, pp. 3-19. http://geodesic.mathdoc.fr/item/TMF_1993_95_1_a0/

[1] Drinfeld V. G., “Quantum groups”, Proc. Int. Congress of Math. (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798–820 | MR

[2] Zamolodchikov A. B., TMF, 65 (1985), 347–359 | MR

[3] Roček M., Phys. Lett., V255 (1991), 554 | DOI | MR

[4] Zachos S. K., “Paradigms of quantum algebras”, Symmetries in science, v. V (Lochau, 1990), Plenum, New York, 1991, 593–609 | DOI | MR

[5] Karasev V. P., Kratkie soobscheniya po fizike, 1988, no. 9, 3 | Zbl

[6] Chaichian M., Elinas D., Kulish P., Phys. Rev. Lett., 65 (1990), 980 | DOI | MR | Zbl

[7] Karasev V. P., J. Sov. Laser Research, 12 (1991), 147 ; препринт No 137, ФИАН, M., 1990 | DOI

[8] Gal'bert O. F., Granovskii Ya. I., Zhedanov A. S., Phys. Lett., A153 (1991), 177 | DOI | MR

[9] Narganes-Quijiano F. J., J. Phys., A24 (1991), 1699

[10] Karassiov V. P., Preprints No 102, 138, FIAN, M., 1991 | MR

[11] Karassiov V. P., Prants S. V., Puzyrevsky V. I., Interaction of electromagnetic field with condensed matter, eds. N. N. Bogolubov, A. S. Shumovsky, V. I. Yukalov, World Sci., Singapore, 1991, 3

[12] Perelomov A. M., Obobschennye kogerentnye sostoyaniya i ikh primeneniya, Nauka, M., 1987 | MR

[13] Weyl H., The theory of groups and quantum mechanics, Dover, N. Y., 1950 | MR

[14] Howe R., “$\theta $-series and invariant theory”, Automorphic forms, representations and $L$-functions, part 1 (Proc. Sympos. Pure Math., Oregon State Univ., 1977), Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, 275–285 | DOI | MR

[15] Milne Thompson L. M., The calculus of finite differences, University Press, Cambridge, 1950 | MR

[16] Katriel J., Solomon A. I., D'Ariano G., Rasetti M., J. Opt. Soc. Am., V4 (1987), 1728 | DOI | MR

[17] Kozierowski M., Mamedov A. A., Chumakov S. M., Phys. Rev., A42 (1990), 1762 | DOI

[18] Katriel J., Solomon A. I., J. Phys., A24 (1991), 2093 | MR | Zbl

[19] Bacry H., J. Math. Phys., 31 (1990), 2061 | DOI | MR | Zbl

[20] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, t. 1, 2, Nauka, M., 1973–1974 | MR

[21] Karasev V. P., Shelepin L. A., Trudy FIAN, 87, 1976, 55 | MR

[22] Holstein T., Primakoff H., Phys. Rev., 58 (1940), 1098 | DOI | Zbl

[23] Brandt R. A., Greenberg O. W., J. Math. Phys., 10 (1969), 1168 | DOI | MR | Zbl

[24] Curtright T. L., Zachos C. K., Phys. Lett., V243 (1990), 273 | DOI | MR

[25] Celeghini E., Rasetti M., Vitiello G., Phys. Rev. Lett., 66 (1991), 2086 | DOI | MR

[26] Infeld L., Hull T. E., Rev. Mod. Phys., 23 (1951), 21 | DOI | MR | Zbl